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Clustering and the Three-Point Function
Yunfeng Jiang, Shota Komatsu, Ivan Kostov, Didina Serban
To cite this version:
Yunfeng Jiang, Shota Komatsu, Ivan Kostov, Didina Serban. Clustering and the Three-Point Func-tion. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2016, 49, pp.454003. �10.1088/1751-8113/49/45/454003�. �cea-01466250�
Clustering and the Three-Point Function
Yunfeng Jianga, Shota Komatsub, Ivan Kostovc1, Didina Serbanc
a Institut f¨ur Theoretische Physik, ETH Z¨urich,
Wolfgang Pauli Strasse 27, CH-8093 Z¨urich, Switzerland
b Perimeter Institute for Theoretical Physics,
Waterloo, Ontario, Canada
c Institut de Physique Th´eorique, DSM, CEA, URA2306 CNRS
Saclay, F-91191 Gif-sur-Yvette, France
jiangyf2008@gmail.com, skomatsu@perimeterinstitute.ca, ivan.kostov & didina.serban@cea.fr
Abstract
We develop analytical methods for computing the structure constant for three heavy operators, starting from the recently proposed hexagon approach. Such a structure constant is a semiclassical object, with the scale set by the inverse length of the operators playing the role of the Planck constant. We reformulate the hexagon expansion in terms of multiple contour integrals and recast it as a sum over clusters generated by the residues of the measure of integration. We test the method on two examples. First, we compute the asymptotic three-point function of heavy fields at any coupling and show the result in the semiclassical limit matches both the string theory computation at strong coupling and the tree-level results obtained before. Second, in the case of one non-BPS and two BPS operators at strong coupling we sum up all wrapping corrections associated with the opposite bridge to the non-trivial operator, or the “bottom” mirror channel. We also give an alternative interpretation of the results in terms of a gas of fermions and show that they can be expressed compactly as an operator-valued super-determinant.
1Associate member of the Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of
Sciences, 72 Tsarigradsko Chauss´ee, 1784 Sofia, Bulgaria
Contents
1 Introduction 2
2 The three-point function and the hexagon proposal 4
2.1 Results and comparison with strong coupling . . . 7
3 Asymptotic structure constant for two BPS and one non-BPS operator 10 3.1 From sum-over-partition to multiple contour integral . . . 11
3.2 Tree-level revisited . . . 13
3.3 Deformation of contours and clustering . . . 15
3.4 The exact result and semiclassical limit . . . 18
3.5 The semiclassical limit for the su(2) sector . . . 20
3.6 The semiclassical limit for the sl(2) sector . . . 21
4 Asymptotic structure constant for three non-BPS fields 22 4.1 Formulation in terms of multiple contour integrals . . . 23
4.2 Taking the semiclassical limit, su(2) . . . 26
4.3 Taking the semiclassical limit, sl(2) . . . 28
5 Bottom mirror excitations 29 5.1 Quantities for bound states at strong coupling . . . 30
5.2 Clustering the mirror particles . . . 36
5.3 The su(2) bottom mirror contribution . . . 39
5.4 The sl(2) bottom mirror contribution . . . 40
6 Fredholm determinants and free fermions 41 6.1 Multiple contour integrals as a Fredholm determinant . . . 43
6.2 The leading term by Fredholm determinant . . . 44
6.3 The leading term as the free energy of a Fermi gas . . . 44
6.4 Semiclassical expansion by nested Fredholm determinant . . . 45
6.5 CFT representation . . . 47
7 Conclusion and outlook 51
A An example for clustering 54
B Scaling limit of the sl(2) and su(2) scalar factors 54 C Computing the phase factor for the asymptotic C123••• 55
D The analytic structure of the transfer matrices in mirror kinematics 58
E The term B3 at strong coupling 61
F The ABJM matrix model and clustering 62
G Separation of variables from clustering 64
1
Introduction
In the strongly-interacting system with a large number of degrees of freedom, it is often the case that the system exhibits emergent collective behaviour, which is entirely different from that of its constituents and provides us with a novel physical picture. The examples of such range from various condensed-matter systems realised in the laboratory, to the AdS/CFT correspondence, which claims that the strongly-coupled CFTs satisfying certain conditions can be described by the gravitational theory in the AdS spacetime.
In this paper, we address one simple but intriguing example of such phenomena in the context of the AdS/CFT correspondence; namely the emergence of the classical string world-sheet from the three-point functions in the planar N = 4 super Yang-Mills theory (SYM). On the one hand, a non-perturbative framework to compute the three-point functions of N = 4 SYM, called the hexagon vertex, was put forward recently in [1]. It describes the three-point functions in terms of the dynamics of “magnons”, which are the elementary fields constituting the gauge-invariant operator. On the other hand, the AdS/CFT implies that the very same object in the strong coupling limit admits a totally different description in terms of the classical string worldsheet and that the three-point function is given by its area [2–5]. However, apart from some partial results given in [1], it is still not clear whether and how these two descriptions are consistent with each other.
these two results. We claim that the semiclassical regime is achieved through a mechanism which we call clustering. When a large number of magnons are put together in the hexagon vertex, they form a sort of bound states, which we call clusters. As we demonstrate in several examples, this clustering phenomenon is essential in order to reproduce the string-theory results from the hexagon vertex. It is worth noting that these clusters bear some resemblance with the bound states in the context of the Thermodynamic Bethe Ansatz.
In order to explain more in detail what we computed with this method, let us briefly recall the structure of the hexagon vertex and the result from the classical string. The hexagon vertex consists of two parts: The one is the asymptotic structure constant which is given by a sum over partitions of the magnons and describes the three-point functions of long operators. The other is the wrapping corrections, which is given by the sums and the integrals of the mirror particles and accounts for the finite size effects. On the other hand, the result from the string theory is given in terms of integrals on the spectral curve, where the integration contours are either around the branch cuts or around the unit circle.
Let us now describe what we achieved in this paper. First we study the asymptotic three-point function of long non-BPS operators in the rank one sectors and show that the result after clustering reproduces the integrals around the branch cuts in the string-theory prediction. Second, in the case of the one non-BPS and two BPS correlators, we sum up the wrapping corrections associated with the edge opposing to the non-BPS operator, taking into account the clustering effect. The result matches nicely with one of the integrals around the unit circle in the string-theory computation.
Our analysis is based upon yet another important observation that, in the regimes of our interest, the expression coming from the hexagon vertex takes the form of the grand-canonical partition function of free fermions. This allows to apply the methods developed in [6] and in [7–9] for the tree-level correlators. When the number of magnons is infinite, these fermions become classical and the result is given by the phase-space integral of this fermion system, which matches the string-theory prediction. This Fermi gas description allows to reproduce the results obtained by clustering in an elegant way, shortcutting the tedious combinatorics. Furthermore, it reveals that the sum over mirror particles on the bottom edge can be nicely re-expressed as the operatorial superdeterminant. However the derivation based on the Fermi gas is not, at the present stage, sufficiently rigorous. Therefore, for the most parts of the paper, we stay on the safe ground of the clustering method and only briefly sketch the Fermi gas approach.
The applicability of these two approaches is not limited to the three-point functions. For instance, the clustering method has proven to be useful for various other problems such as the strong coupling limit of the scattering amplitudes in N = 4 SYM [10], which
was otherwise obtained by different methods [11, 12]. Clustering-like methods were used to compute the partition functions in N = 2 gauge theories in the Nekrasov-Shatashvilli limit [13–15], and the integrable models describing non-equilibrium processes [16–18]. On the other hand, the Fermi gas approach is used extensively to study the M-theoretic large N limit of the ABJM and related theories as well as the super-conformal index in four dimensions [19, 20]. Our analysis indicates that these approaches are deeply connected.
The rest of the paper is structured as follows. In section2we review the computation of the three-point function and the hexagon vertex and summarise our results, as well as the string-theory prediction at strong coupling [21]. Then in section3 we study the asymptotic structure constant for the three-point function of one non-BPS and two BPS operators. For this purpose, we first re-express the sum-over-partitions formula in the hexagon proposal as a multiple contour integral. We then explain the basic idea of the clustering using the tree-level example and show that the method can be applied at finite coupling. Next, in section4we generalise it to the case of three non-BPS fields and reproduce the string-theory prediction. In section5, we turn to the wrapping corrections and summarise expressions for the basic quantities at strong coupling. Using such expressions, we analyse the clustering of the mirror particles and obtain the expression consistent with the string theory. Lastly in section 6, we show that these results can be computed alternatively using the Fermi gas approach and the Fredholm determinant. We in particular show that the summation over the mirror particles can be expressed as the generalised Fredholm determinant2, which can be further converted into an operator-valued superdeterminant. We conclude in section
7. Several appendices are provided in order to explain technical details and elucidate the relation between the clustering and other methods: In Appendix F, we study the ABJM matrix model using the clustering method, and in AppendixG, we relate the hexagon vertex and the separation of variables at tree-level using the clustering.
2
The three-point function and the hexagon proposal
The three-point function of operators in the N = 4 planar SYM theory is fixed up to a constant by the conformal invariance,
hO1(x1)O2(x2)O3(x3)i =
C123(g)
|x12|∆12|x13|∆13|x23|∆23
, (2.1)
with xi vectors in the 3 + 1 dimensional Minkowski space, ∆i the conformal dimension of
the operator Oi and ∆ij = ∆i+ ∆j− ∆k. The constant C123 is given in terms of the initial
data of the three operators, namely the charges of the global symmetry group P SU (2, 2|4) and the charges of the infinite symmetry group associated to integrability. The latter ones, dependent on the coupling constant g, can be encapsulated, at least in the regime of in the small g, by three collections of rapidities u1, u2, u3, each associated to one of the operators
O1(x1), O2(x2), O3(x3). At g = 0 the three sets of rapidities are determined by Bethe
ansatz equations for three P SU (2, 2|4) spin chains with lengths L1, L2 and L3. At non-zero
values of the coupling constant g, the spin chains acquire long-range interaction and the so-called asymptotic Bethe ansatz is not exact anymore. The long-range corrections can be interpreted as coming from virtual particles circulating in the so-called mirror channel, where time and space are interchanged. These virtual particles are called mirror particle. Their contribution to the spectrum of conformal dimensions ∆(g) can be exactly determined via a set of functional equations known under the name of Quantum Spectral Curve, equivalent to a system of Thermodynamic Bethe Ansatz equations. In the large volume limit the contribution of the virtual particles is exponentially small.
Through the AdS/CFT correspondence [23], the point function is dual to a three-string interaction connecting three three-strings with energies ∆1, ∆2, ∆3. The rapidities can
be then associated to the momenta of excitation modes, or magnons, propagating on the 1+1 dimensional worldsheet. For a particular subset of the operators, the BPS operators, the conformal dimensions do not depend on the coupling constant g and the associated rapidities are trivial (i.e. infinite). We are going to use a bullet to symbolise a non-BPS operator and an empty circle to denote the BPS one with the same global charges. To remove some trivial combinatorial factors we are dividing the three-point function by the three-point function of the corresponding BPS operators, e.g.
C123••◦ ≡ C
••◦ 123
C123◦◦◦pN1N2 (2.2) denotes the three-point function of two non-BPS and one BPS operator. In the above formula, √Ni are the normalisation of the three incoming states, which can be expressed
in terms of the Gaudin determinants. In this work we are not considering the explicit expression of the norms, and prefer considering the unnormalised structure constants C123
defined in (2.2) instead of the normalised structure constants C123. The semiclassical limit
of the norms in the absence of mirror correction was taken in [7, 24].
An all-loop prescription to compute the three-point function was given in [1]. The guiding principle of the proposal is to split the worldsheet of the three interacting strings into two overlapping hexagons, and then sum over all possible ways of distributing the magnon excitations between the two hexagons, u1 = α1 ∪ ¯α1, u2 = α2 ∪ ¯α2, u3 = α3 ∪ ¯α3
answer is
1 11
2
2 3 3
Figure 2.1: A possible arrangement of excitations for the hexagon form factors.
[C123•••]asympt = X αi∪ ¯αi=ui 3 Y i=1 (−1)|α1|+|α2|+|α3| w `31(α1, ¯α1) w`12(α2, ¯α2) w`23(α3, ¯α3) × H(α1|α3|α2)H( ¯α2| ¯α3| ¯α1) . (2.3)
Explicit expressions for transition factors w`i−1,i(αi, ¯αi) and hexagon form factors H(α1|α3|α2)
were proposed in [1] and will be given later. The building blocks of the hexagon form factors are the bi-local hexagon amplitudes h(u, v) proposed in [25] and the elements of the Beisert’s scattering matrix [26]. Here we are going to consider only structure constants of operators from the rank-one sectors su(2) and sl(2) and we are therefore not going to use the matrix structure of the hexagon form factors.
YY Y Y Y Y Y_ Y_ Y_ Y_ Y_ Y_ Y_ Y
l
B 3L
1L
2L
2L
+L
3-L
1 = Rl
=L
1+L
3-L
2 Rl
=L
1+L
2-L
3Figure 2.2: Vacua and su(2) excitations in the reservoir picture of BKV [1].
To connect with the weak-coupling picture and the corresponding notations, it is useful to represent the three-point function we consider in the reservoir picture of [1] represented
in2.2. In this picture, the first operator O1 is of the form Tr(ZL1−M1YM1) + . . . , the second
operator O2 is of the form Tr( ¯ZL2−M2Y¯M2) + . . . , and the third operator O3, the reservoir,
is built as Tr(Z + ¯Z + Y − ¯Y )L3−M3( ¯Y − ¯Z)M3 + . . .. This type of structure constant is
called type I-I-II in [27], since two operators belong to the “left” su(2) sector and one belongs to the “right” su(2) sector in the sense that the operator O2 can be obtained from
Tr(ZL2YM2) + . . . and O
3 from Tr(ZL3−M3Y¯M3) + . . . by one of the twisted translation
defined in [28] and used in [1]. A similar definition works in the sl(2) sector.
The inclusion of wrapping corrections to equation (2.3) is done by including an infinite tower of excitations, as well as their bound states, circulating in the three mirror channels denoted by black edges in figure2.1. The summation is done over their rapidities and their polarisations. The general expression is too complicated to be reproduced here; instead, we can illustrate the type of contribution on the case of a single non-BPS operator. We consider only the mirror particles in the channel opposed to that operator, as showed in figure 2.3. Following [25] we call this channel the bottom channel. In this case, the asymptotic and mirror contributions conveniently factorise,
C•◦◦= [C•◦◦]bottom[C•◦◦]asympt. (2.4) Schematically, given in terms of only the fundamental excitations, the expression of the wrapping corrections is given by [1, 25]
[C•◦◦]bottom= Z ∞
−∞
dw µ(wγ) eip(wγ)`BT (wγ) h6=(wγ, wγ) h(u, w−3γ) , (2.5)
with `B = 12(L2 + L3 − L1) the length of the bottom bridge of the correlator, opposed to
the operator O1 and T (w) the su(2|2) spin chain transfer matrix [26]. The full result takes
into account all the bound states and will be given in the corresponding section. Here and below the index γ stands for the mirror transformation and we use the shorthand notations
h(u, v) ≡Y i,j h(ui, vj) , h6=(u, u) = Y i6=j h(ui, uj) . (2.6)
2.1
Results and comparison with strong coupling
In the case when the incoming operators correspond to semiclassical strings, the lengths L1, L2, L3 of the three chains and the numbers of the magnon excitations M1, M2, M3 are
large. The semiclassical limit is controlled by a small parameter such that Li and Mi
remain finite when → 0. This limit exists for any value of the ’t Hooft coupling g. In addition to the semiclassical limit, one can take the strong coupling limit where the effective
bottom mirror excitations
physical excitations
Figure 2.3: The physical and bottom mirror excitations.
coupling g0 = g remains finite when → 0. Based on the experience with the spectrum [29], we may expect that, for sl(2), the results for the semiclassical strings can be applied safely to small values of Li.
The summation over the different ways of partitioning the rapidities in equation (2.3), as well as the summation over the mirror particles remains an open problem in general. Here we report some modest progress in taking the sum and the semiclassical limit in three particular cases when the operators belong to the rank-one sectors su(2) and sl(2):
• the expression of the asymptotic part of the structure constant for one non-BPS and two BPS operators, [C•◦◦]asympt for any value of the coupling constant,
• the expression of the asymptotic part of the I-I-II structure constant3 for three non-BPS operators belonging to two different su(2) or sl(2) sectors, [C•••]asympt, for any
value of the coupling constant,
• the expression of the bottom mirror contribution for one non-BPS and two BPS operators, [C•◦◦]bottom in the strong coupling limit.
The first case is a relatively simple generalisation of the result obtained by [6,7,9] at tree-level. Here we use a slightly different method of taking the semiclassical limit, based on an integral representation of the sums in (2.3) which has already appeared in [9]. This method is alternative to the Fredholm determinant method used there and it is easily adaptable to situation when the structure constant cannot be written exactly as a determinant. Finally, the structure of the integrals in the third case ressemble strongly that from the first two
cases, and we are able to take the sum over bound states exactly in the strong coupling limit.
The answer for the semiclassical structure constants is given in terms of quasi-momenta associated to the three operators, which encode the corresponding rapidities. For operators duals to semiclassical strings, the rapidities are distributed on a set of cuts, which connect different sheets of the quasi-momenta. We are going do denote by ˜p(k) the sphere part and
by ˆp(k) the AdS part of the quasi-momentum associated to the operator O
k. The definition
of the quasi-momenta will be given in the main text. The results for the su(2) and sl(2) sectors are
log[C123•••]asymptsu(2) = −1 I Cu1∪u2 du 2π Li2 h ei ˜p(1)L +i ˜p (2) L −i˜p (3) R i − 1 I Cu3 du 2π Li2 h ei ˜p(3)R +i ˜p (2) L −i˜p (1) L i , (2.7) log[C123•••]asymptsl(2) = 1 I Cu1∪u2 du 2π Li2 h ei ˆp(1)L +i ˆp (2) L −iˆp (3) R i + 1 I Cu3 du 2π Li2 h ei ˆp(3)R +i ˆp (2) L −iˆp (1) L i . (2.8) where Cuk is a contour encircling counterclockwise the support of the rapidities uk. The
result for [C123•◦◦]asympt is the particular case where u
2 = u3 = ∅. We would like to emphasise
that the expression above are valid when the length of the three operators L1, L2 and L3 are
large and the supports of u1, u2 and u3 are well separated. The so-called heavy-heavy-light
diagonal limit, when the length of one of the operators, say L3, is small and in addition
u1 = u2 was studied in [30, 31].
A surprisingly similar form is taken by the result of the resummation of the virtual particles. Here we succeeded to take the sum only of the mirror particles for the structure constant with one non-BPS operator in the channel opposed to the non-trivial operator,
log[C123•◦◦]bottomsu(2) = 1 I U du 2π Li2 h ei( ˆp(2)+ ˆp(3)−ˆp(1))i− Li2 h ei( ˜p(2)+ ˜p(3)−˜p(1)(x))i , (2.9) log[C123•◦◦]bottomsl(2) = −1 I U du 2π Li2 h ei( ˆp(2)+ ˆp(3)−ˆp(1)) i − Li2 h ei( ˜p(2)+ ˜p(3)−˜p(1)(x)) i , (2.10) with the contour of integration U encircling now the Zhukovsky cut with u between −2g and 2g.
The three-point functions at strong coupling admit a completely different description, namely in terms of the area of the classical string worldsheet. The computation from the string theory side was completed recently building on earlier works [21]. In both su(2) and sl(2) sectors, the result is composed of three terms,
log C123••• = log[C123•••]asympt+ log[C123•••]wrapping+ Norm . (2.11) For the type I-I-II three-point functions in the su(2) sector, the asymptotic part and the
wrapping part are given on the string theory side by4 log[C123•••]asymptsu(2) = −1
I Cu1∪u2 du 2πLi2 h ei ˜p(1)L +i ˜p (2) L −i˜p (3) R i − 1 I Cu3 du 2πLi2 h ei ˜p(3)R +i ˜p (2) L −i˜p (1) L i , (2.12)
log[C123•••]wrappingsu(2) = 1 I U du 2π Li2 h ei( ˆp(1)+ ˆp(2)−ˆp(3)) i − Li2 h ei( ˜p(1)L + ˜p (2) L −˜p (3) R ) i (2.13) +1 I U du 2π Li2 h ei( ˆp(2)+ ˆp(3)−ˆp(1))i− Li2 h ei( ˜p(2)L + ˜p (3) R −˜p (1) L ) i +1 I U du 2π Li2 h ei( ˆp(3)+ ˆp(1)−ˆp(2))i− Li2 h ei( ˜p(3)R + ˜p (1) L −˜p (2) L ) i +1 I U du 2π Li2 h ei( ˆp(3)+ ˆp(1)+ ˆp(2))i− Li2 h ei( ˜p(3)R + ˜p (1) L + ˜p (2) L ) i .
As is clear from the above expressions, [C123•••]asymptprecisely matches the result of our analysis
(2.7) and (2.8). Furthermore, when restricting to the one non-BPS and two BPS correlators, we can see that the first term in [C123•••]wrapping coincides with our result of the resummation
of the bottom mirror particles [C123•◦◦]bottom in (2.10) and (6.41). Similar match can be seen
also in the sl(2) sector, where the result from the string theory reads log[C123•••]asymptsl(2) =1 I Cu1∪u2 du 2πLi2 h ei ˆp(1)L +i ˆp (2) L −iˆp (3) R i +1 I Cu3 du 2πLi2 h ei ˆp(3)R +i ˆp (2) L −iˆp (1) L i , (2.14) log[C123•••]wrappingsl(2) =1 I U du 2π Li2 h ei( ˆp(1)L + ˆp (2) L −ˆp (3) R ) i − Li2 h ei( ˜p(1)+ ˜p(2)−˜p(3)) i (2.15) +1 I U du 2π Li2 h ei( ˆp(2)L + ˆp (3) R −ˆp (1) L ) i − Li2 h ei( ˜p(2)+ ˜p(3)−˜p(1))i +1 I U du 2π Li2 h ei( ˆp(3)R + ˆp (1) L −ˆp (2) L ) i − Li2 h ei( ˜p(3)+ ˜p(1)−˜p(2))i +1 I U du 2π Li2 h ei( ˆp(3)R + ˆp (1) L + ˆp (2) L ) i − Li2 h ei( ˜p(3)+ ˜p(1)+ ˜p(2))i .
The remaining factors in [C123•••]wrapping supposedly come from other mirror channels. It would be an important future problem to reproduce those remaining terms by resumming the mirror particles in other channels.
3
Asymptotic structure constant for two BPS and one
non-BPS operator
In this section we are computing the structure constant for the case of a single non-BPS operator. Although this can be considered as a particular case of the one treated in the
next section, we prefer to work out in detail the clustering method on the simpler case, and then have a result ready to use for to the more complicated case. Since the su(2) and sl(2) sectors are largely similar, we treat only the former in detail, and just give the results and point out the main difference for the latter.
3.1
From sum-over-partition to multiple contour integral
In the definition of the structure constant, the three operators are represented by on-shell states of three different spin chains of lengths L1, L2, L3. Only the first chain of length
L ≡ L1 has non-trivial excitations (magnons) with momenta p1, . . . , pM, M ≡ M1. The
momenta are parametrised by the corresponding rapidities u = {u1, . . . , uM} according to
eip(u) = x(u + i/2)
x(u − i/2). (3.1)
Above, we have rescaled the rapidity variables by which will be set at the typical value for the rapidities u. In the regime dual to semiclassical strings, this overall scale is ∼ 1/L1.
The semiclassical limit is → 0. The Zhukovsky variable x(u) is defined as
x(u) = u + q
u2− (2g)2
2g . (3.2)
The rapidities u satisfy the Bethe equations
eiφj = 1 , j = 1, . . . , M, (3.3) where φj is the total scattering phase for the j-th magnon
eφj = e−ip(uj)L Y
k(6=j)
S(uj, uk), (3.4)
S(u, v) being the scattering matrix, which can be represented as the ratio S(u, v) = h(v, u)
h(u, v). (3.5)
The function h(u, v), which is given in our case by h(u, v)su(2) ≡ hY Y(u, v), is the building
block for the hexagon expansion in the configuration described above. It is given by the product of three factors,
h(u, v)su(2) =
u − v u − v + i
1
where s(u, v) is the symmetric part,
s(u, v) = (1 − 1/x
+y+) (1 − 1/x−y−)
(1 − 1/x+y−) (1 − 1/x−y+) (3.7)
and σ(u, v) = 1/σ(v, u) is the square root of the BES dressing phase [32,33]. The reason to split h(u, v) as above is that at tree (g = 0) level, s(u, v) = σ(u, v) = 1. It will be important in the following that neither s(u, v) nor σ(u, v) has singularities close to u = v. We use the notation x± = x(u ± i/2) and y± = x(v ± i/2). The unnormalised structure constant, is defined as a sum over partitions of the rapidities u into two subsets, u = α ∪ ¯α,
[C123•◦◦]asympt≡A = X α∪ ¯α=u (−1)| ¯α| Y j∈α eip(uj)`R Y j∈α,k∈ ¯α 1 h(uk, uj) , (3.8)
where `R = 12(L1 + L3− L2) is the length of the bridge between the first operator (on the
top) and the third one. In order to have a complete match with the original tree-level result reported in [6, 24], we will work with an equivalent representation,
A = X α∪ ¯α=u (−1)|α| Y j∈α e−ip(uj))` Y j∈α,k∈ ¯α 1 h(uj, uk) , ` ≡ `L, (3.9)
where `L= 12(L2+ L1− L3) is the length of the bridge connecting the first and the second
operator. The equivalence of the two expression can be shown by using the Bethe ansatz equations (3.3) with L1 = `L + `R. Formally, at tree-level, the two expressions (3.9) and
(3.8) can be obtained from each other by exchanging `L and `R and sending → −.
Extending the tree-level observation in [9], the sum over partitions (3.9) can be written as a multiple contour integral
A = N X n=0 1 n! I Cu n Y j=1 dzj 2π F (zj) n Y j<k h(zj, zk) h(zk, zj) , (3.10)
where the integration contour Cu closely encircles the rapidities u = {u1. . . uN}
counter-clockwise, the function F (x) is given by
F (z) = e −ip(z)` µ(z) h(z, u) , h(z, u) ≡ N Y j=1 h(z, uj) , (3.11)
and the measure
µ(z) = (1 − 1/x
+x−)2
(1 − 1/(x+)2)(1 − 1/(x−)2) (3.12)
In the all loop pairwise interaction
∆all(zj, zk) ≡ h(zj, zk) h(zk, zj) = ∆(zj, zk) s(zj, zk)2 (3.13)
the dressing factor drops out due to the anti-symmetry of the dressing phase. In the semiclassical limit → 0, the deviation of the interaction ∆all(u, v) with respect with its
tree level value ∆(u, v) is subleading,
∆all(u, v) = ∆(u, v) 1 − c(u, v, g)22+ O(3) , (3.14) where c(u, v, g) is some function of the rapidities u and v and the effective coupling g0 = g. It is important that even at strong coupling, where g0 is finite, the correction to the interaction is subleading. A similar property is valid for the measure µ(u)
µ(u) = 1 − c(u, u, g) 2+ O(3) . (3.15) This will allow us to take the semiclassical limit of the asymptotic contribution for any value of the coupling constant, including strong coupling. The main steps of the derivation can be understood on the tree-level example, which can be treated exactly and will be worked out in detail in the following. The clustering procedure explained below works exactly as in the tree-level, as long as the integration contours are kept as distance from the cuts of the dressing phase σ(u, v), that is out of the region −2g0 < Re(zk) < 2g0. This is certainly the
case for semiclassical strings.
3.2
Tree-level revisited
The structure constant of one non-BPS and two BPS operators at three level A was first studied thoroughly in [24] and [6]. In this section, we revisit the tree-level result by a different method which allows direct generalisations to all loops.
The starting point is the multiple integral contour integral (3.10)
A = N X n=0 1 n! I Cu n Y j=1 dzj 2π F (zj) n Y j<k ∆(zj, zk), (3.16)
where the different ingredients take their tree-level values5
F (z) = e
−ip(z)`
h(z, u), h(z, u) =
z − u
z − u + i . (3.17)
The sum (3.16) is given by single integrals coupled by the pairwise interaction ∆(u, v) defined as ∆(u, v) =h(u, v) h(v, u) = (u − v) 2 (u − v)2+ 2 (3.18) = 1 + i/2 u − v − i − i/2 u − v + i.
In other words, the function ∆(u, v) differs from 1 only when |u − v| ∼ , and it has two poles at u = v ± i. For later convenience, we define a generalisation of this function
∆mn(u, v) = (u − v)(u − v + i(m − n)) (u − v + im)(u − v − in) (3.19) = 1 − mn m + n i u − v + im + i u − v − in ,
so that ∆(u, v) = ∆1,1(u, v). The summation limit in (3.16) can be extended to infinity,
since the result of integration is zero if there are more integrals than rapidities in the set u. The multiple contour integral representation (3.16) is our starting point. Similar integrals have appeared recently in the context of integrable probabilities for example in [16,17] and [18].
Semiclassical limit. The rhs of (3.16) can be viewed as a grand canonical partition function of a matrix model. This matrix model appeared when computing the partition function of dimensionally reduced SYM with four supercharges [34]. The semiclassical limit (large N or large chemical potential for the grand canonical partition function) was found in [35] using the standard matrix model techniques. The spectral curve of the matrix model is associated with an elliptic Riemann surface with two parallel cuts at distance from each other.
The semiclassical limit we are interested in is more subtle. It consists of taking the limit L, M → ∞ so that M/L ∼ 1, or taking → 0 so that M remains finite.6 In this limit, which is very similar to the Nekrasov-Shatashvili limit [13], the standard matrix model techniques do not work. The leading and the subleading term of the partition function were evaluated in [9] by representing the partition function as a Fredholm determinant and resolving the corresponding Riemann-Hilbert problem. A shorter, although less rigorous derivation used the mapping to a system of chiral fermions.
6This is the large L limit of a solution of the Bethe equations, characterised by one or more Bethe strings
with mode numbers nk and filling fractions αk = Mk/L1. When L1→ ∞, the distribution of the magnon
rapidities along each Bethe string converges to a continuous linear density. This limit of the spin chain has been first studied in [36] and then rediscovered in the context of AdS/CFT in [37].
Here we will give a rigorous derivation of the semiclassical limit based on an exact evaluation of each term in the sum in (3.16) and then taking the limit. We will observe a formation of bound states in close analogy to the bound states of instantons appearing in the Nekrasov-Shatashvili limit.
3.3
Deformation of contours and clustering
Here we will set up a procedure which allows to perform an expansion in the parameter around the semiclassical limit → 0, L ∼ M ∼ 1/ → ∞ of the functional A in (3.16). Namely, we deform the integration contours sequentially so that they become widely separated and far way from the support of u, as is shown in figure 3.4. After the contour
Figure 3.4: Deformation of the integration contours. Here Ck is the deformed contour of
the integration variable xk, which is situated at a distance larger than from all the other
contours.
deformation, we have |zj−zk| and the singularities in the multiple integrals are removed.
In the procedure of deformation of contours, one has to take into account the residues of the poles in the interaction terms ∆(zi, zj) in (3.16). This leads to a phenomenon we call
clustering which was considered in various forms in [13, 34], in [14, 15] and in [16, 18] and which is reminiscent of the formation of bound states as solutions of the Bethe equations. A similar procedure was suggested in [10] in order to take the strong coupling limit of the scattering amplitudes for gluons. Let us consider in more detail an integral of the type
In= I Cu n Y j=1 dzj 2π F (zj) n Y j<k ∆(zj, zk), (3.20)
which corresponds to the n-th term in the sum in (3.16). The integrand is a product of functions F (z) and ∆(zi, zj). We can imagine a collection of n particles, each particle
zi is associated with a function F (zi) and between any two particles zi and zj, there are
interactions described by the function ∆(zi, zj). Then the integrand can be represented by
the diagram shown in figure3.5.
Figure 3.5: A diagrammatic representation of the integrand of In.
In order to illustrate the idea, we analyse an example for n = 3 explicitly. We start with I3 =
I
Cu
dz1dz2dz3
(2π)3 F (z1)F (z2)F (z3)∆(z1, z2)∆(z1, z3)∆(z2, z3) (3.21)
We first deform the contour of integration for z3 from Cu to a contour C3 which is situated
outside Cu at a distance larger than . There are poles at z3 = z2± i and z1 ± i due to
the interaction ∆(z1, z3) and ∆(z2, z3), respectively. If we take the pole at z3 = z2− i, the
residue is proportional to the following integral I Cu dz1dz2 (2π)2F (z1)F (z2)F (z2− i)∆(z1, z2)∆(z1, z2− i). (3.22) Because F (z2)F (z2− i) = z2− 3i/2 z2 + i/2 ` z2− u + i z2− u − i (3.23) is analytic inside the contour Cu, the integration over z2 gives zero. The same argument
works for z3 = z1− i. This implies that we only need to consider the poles z3 = z2+ i and
z3 = z1+ i. If we take the pole z3 = z2+ i, the result reads
1 2 I Cu dz1dz2 (2π)2F (z1)F (z2)F (z2+ i)∆(z1, z2)∆(z1, z2+ i) . (3.24)
We have taken here into account that, while deforming the counter-clockwise contour Cu
into Ck, the contours surrounding the poles will be oriented clockwise. Let us define the
functions F1, F2, F3, etc. by
Using the fact that
∆(u, v)∆(u, v + i) = ∆1,2(u, v) (3.26)
where ∆1,2(u, v) is defined in (3.19), we can write the residue (3.24) as
1 2 I Cu dz1dz2 (2π)2F (z1)F2(z2)∆1,2(z1, z2). (3.27)
This result can be interpreted as the following. Taking the residue gives rise to a cluster, or bound state, of length 2. The function associated to this cluster is given by F2(z) and
its interaction with a fundamental particle at the point z0 is described by ∆1,2(z, z0). This
is symbolised graphically in figure3.6, left.
Figure 3.6: The clustering of fundamental particles into bound states.
When moving their integration contours from Cu to Cj, the bound states themselves
undergo further clustering and form larger bound states. A length n bound state is as-sociated to the wave function Fn(z) defined in (3.25) and the interaction between bound
states of length m and n is described by ∆mn(z, w). The full result of our example n = 3,
which illustrates the origin of the combinatorial factors, is given in appendixA. In terms of diagrams, it is given in figure3.7.
As we can see, the result is given by the sum of all possible bound states, each bound state of length n multiplied by a factor 1/n. To see that this is true in general, let us consider the integral with a bound state of length m and length n
I Cu dzj 2π dzk 2π· · · Fm(zj) m × Fn(zk) n ∆mn(zj, zk) · · · (3.28) Suppose we now want to deform the contour zk to Ck and pick the pole zk = zj+ im. The
Figure 3.7: The final result of I3after deforming the contours. Here the black dots mean the
integration contour for xj is Cj. The numbers in blue represent the multiplicities of clusters
and they are given by equation 3.35, for example C31,1,1= 1, C31,2= 3 and C33= 2.
extra contribution from the pole is I Cu dzj 2π· · · Fm(zj) m × Fn(zj + im) n × mn m + n · · · (3.29) = I Cu dzj 2π· · · Fm+n(zj) m + n
where we have used that
Fm(z)Fn(z + im) = Fm+n(z), Res
v=u+im∆mn(v, u) =
imn
m + n. (3.30) In what follows we will denote the fusion rules like (3.29) simply as
Fm(zj) m × Fn(zj + im) n → Fm+n(zj) m + n . (3.31)
The fusion rules ensure that the final result is a sum over all possible bound state configu-rations. Each configuration comes with a combinatorial factor. We will derive these factors and write down an exact expression for In in the next section.
3.4
The exact result and semiclassical limit
In this section, we give an exact expression for In and A and then take its semiclassical
limit. As discussed above, while deforming the contour we need to pick up poles which lead to the formation of bound states. The final result is a sum over all possible configurations of bound states In = n X k=1 X q1+···qk=n Cnq1,··· ,qk k Y j=1 I Cj dzj 2π Fqj(zj) qj n Y i<j ∆qi,qj(zi, zj). (3.32)
Here k is the number of bound states in a given configuration and q1 ≤ · · · ≤ qk are the
for the bound state defined in (3.25) and ∆qi,qj(zi, zj) is defined in (3.19). The combinatorial
factor Cq1,··· ,qk
n counts the number of the bound state configuration with lengths {q1, · · · , qk}.
In what follows, it is convenient to represent the bound state configuration in a different way. Suppose among the bound state configurations {q1, q2, · · · , qk}, dl of them have length
l (l = 1, 2, · · · ), then we can represent the configuration by a vector ~d = {d1, d2, · · · }:
{q1, · · · , qk} = {1 · · · 1 | {z } d1 , · · · , l, · · · , l | {z } dl , · · · } 7→ ~d = {d1, d2, · · · }. (3.33)
We will use the two notations interchangeably. The following two obvious identities will be useful k X j=1 F (qj) = ∞ X l=1 dlF (l), k Y j=1 F (qj) = ∞ Y l=1 F (l)dl. (3.34)
In particular, the constraintPk
j=1qj = n can be rewritten as P ldll = n. We have Cq1,...,qk n = 1 d1!d2! · · · n q1 n − q1 q2 · · ·qk qk (q1− 1)! · · · (qk− 1)! (3.35) = 1 d1!d2! · · · n! q1· · · qk = Qn! lldldl! .
Let us explain briefly how to obtain the first line of the expression above. It is constituted from three different blocks: the middle one is the way to make k clusters of lengths n1 ≤
. . . ≤ nk out of n variables, while the first block insures that clusters with the same number
of elements are indistinguishable. The last block gives the number of different ways to arrange the objects inside each cluster. For a cluster with n1 elements, one can choose the
label of the surviving integration variable at will, while the number of different possible orders of clustering for the other variables is (n1 − 1)!. Inserting (3.35) into (3.32) and
summing over n we obtain the exact result A = X k X q1≤···≤qk 1 d1!d2! · · · k Y j=1 I Cj dzj 2π Fqj(zj) q2 j k Y i<j ∆qi,qj(zi, zj) (3.36) =X k 1 k! X q1,... ,qk k Y j=1 I Cj dzj 2π Fqj(zj) q2 j k Y i<j ∆qi,qj(zi, zj) .
In the last line the summation over qj is unrestricted. This exact expression can be taken
as the starting point for a systematic semiclassical expansion. There are two sources of corrections, from the wavefunction Fn(z) and from the interaction ∆mn(zi, zj),
Fn(z) = F (z)n+ n(n − 1) 2 F (z) n−1∂ zF (z) + O(2) (3.37) ∆mn(zi, zj) = 1 − mn (zi− zj)2 2+ O(3) .
If we are interested in the leading order of expansion of (3.36) we can replace Fn(z)
by Fn(z) and ∆
mn(zi, zj) by 1, which simplifies (3.36) drastically. The multiple integrals
decouple and the result exponentiates, A ' X k 1 k! k Y j=1 X qj I Cj dzj 2π F (zj)qj q2 j = exp I Cu dz 2π X q F (z)q q2 . (3.38)
Here the integration contour is far way from the support of u, but now we can deform it back to encircle closely the support of the rapidities u. We recognise in the expression above the expansion of the dilogarithm. Taking into account the subleading corrections from (3.37) we obtain the first two terms from [9]
logA = I Cu dz 2πLi2[F (z)] − 1 2 I C×2u dzdz0 (2π)2 log [1 − F (z)] log [1 − F (z0)] (z − z0)2 + . . . . (3.39)
To avoid the singularity when z and z0 coincide in the double contour integration above, the two contours can be separated, which is equivalent to taking the principal value integral. More terms in the expansion (3.39) can in principle be obtained by a cluster expansion of (3.36).
3.5
The semiclassical limit for the su(2) sector
We now specialise the expression in (3.39) to the particular case of the su(2) sector
logA ' I Cu dz 2πLi2e −ip(z)`+iGu(z) , (3.40)
which agrees with the results in [6] and [8]. Above, we denoted with p(z) and Gu(z) the
momentum and resolvent at tree level, in the semiclassical limit → 0,
ptree(z) = z , G tree u (z) = N X i=1 z − ui . (3.41)
The all-loop result has exactly the same structure, but with the quasi-momentum re-placed by its full expression, which contains now the dressing phase,
Gu(z) = N X i=1 z − ui − i log σ(z, ui) . (3.42)
In the physical regime the dressing phase can be expressed as
−i log σ(u, v) = χ(u+, v−) + χ(u−, v+) − χ(u+, v+) − χ(u−, v−) ' 2∂
with χ(u, v) given by an integral representation [38]. Defining the sphere all-loop quasi-momenta ˜p(k)(z) by ˜ p(2,3)(z) = x 0(z)L 2,3 2x(z) , p˜ (1) (z) = x 0(z)L 1 2x(z) − Gu(z) , (3.44) the semiclassical limit of the asymptotic all-order contribution is given by
logA = I Cu dz 2πLi2 h ei( ˜p(3)(z)− ˜p(2)(z)− ˜p(1)(z))i= − I Cu dz 2πLi2 h ei( ˜p(3)(z)− ˜p(2)(z)+ ˜p(1)(z))i (3.45)
The last expression is the semiclassical limit of (3.9). We used that
(e−i˜p(1)(z))on the first sheet = (ei ˜p (1)(z)
)on the second sheet (3.46)
which is a consequence of the classical limit of the Bethe equations (3.4),
˜
p(u + i0) + ˜p(u − i0) = Lp(u) − (Gu(u + i0) + Gu(u − i0)) = 0 mod(2π), (3.47)
and that the contour of integration changes its orientation when deformed to the second sheet.
In the strong coupling limit the dressing phase simplifies, χ(u, v) ' u−v log1 −xy1 . Since u−v1 = −∂u∂v(u − v) log(u − v) the resolvent becomes
Gu(z) = N X i=1 x0(ui) x(z) − x(ui) − p(x) N X i=1 x0(ui) x2(u i) ≡ Gu(x) − ∆ − L 2 p(x) , (3.48) with ∆ − L the anomalous dimension, or the spin-chain energy. The quasi-momenta ˜p(k)(z)
assume in this limit the simpler form ˜ p(2,3)(z) = x 0(z)L 2,3 2x(z) , p˜ (1) (z) = x 0(z)∆ 2x(z) − Gu(x(z)) . (3.49)
3.6
The semiclassical limit for the sl(2) sector
The expression for the three-point function with a single non-BPS operator in the sl(2) sector is the same as (3.9) with h(u, v) = hsu(2)(u, v) replaced with the corresponding sl(2)
quantity hsl(2)(u, v) = x+− y− x−− y+hsu(2)(u, v) = u − v u − v − i 1 − 1/x−y+ 1 − 1/x−y− 1 − 1/x−y+ 1 − 1/x+y+ 1 σ(u, v). (3.50)
At tree-level, Asl(2) can be obtained from Asu(2) just by sending → − and ` → −`. At
higher loop, the change comes from changing the expression of h(u, v) as in (3.50), which affects the expression of the quasi-momenta in the semiclassical limit,
fsl(2)(u) → ei( ˆp
(3)(z)− ˆp(2)(z)− ˆp(1)(z))
. (3.51)
The quasi-momenta appearing in the asymptotic part of the sl(2) structure constant corre-spond now to the AdS part of the spectral curve [39, 40],
ˆ p(2,3)(z) = x 0(z)L 2,3 2x(z) , pˆ (1) (z) = x 0(z)L 1 2x(z) + Gu(x(z)) . (3.52) The slightly different appearance of (3.52) with respect to (3.49) is due to the extra factor in the second member of (3.50).
We can therefore write the semiclassical limit of the asymptotic all-order contribution in the sl(2) sector as logAsl(2) = − I Cu dz 2πLi2 h ei( ˆp(3)(z)− ˆp(2)(z)− ˆp(1)(z))i = I Cu dz 2πLi2 h ei( ˆp(3)(z)− ˆp(2)(z)+ ˆp(1)(z))i . (3.53) Upon permutation of indices 2 and 3, which is possible due to symmetry, this expression coincides with the strong coupling result (2.14).
4
Asymptotic structure constant for three non-BPS
fields
Here we consider the all-loop prediction for a configuration equivalent to that studied in [24] where two of the operators belong to the left sector and the third operator belongs to the right sector of so(4) = su(2)L⊕ su(2)R. The excitations for the three operators are chosen
to be the longitudinal scalars
O1 ∈ su(2)L: vacuum Z
L1, M
1 excitations Y = Φ1 ˙2,
O2 ∈ su(2)L: vacuum ZL2, M2 excitations Y = Φ1 ˙2,
O3 ∈ su(2)R: vacuum Z
L3, M
3 excitations ¯Y = Φ2 ˙1.
After the twisted rotation the three operators are mapped to operators type {Z, Y } at the origin, { ¯Z, ¯Y } at infinity, and { ˜Z, ˜Y } at some finite point, say ~x = (0, 0, 1, 0), where
˜
Z = 12 Z + Y + ¯Z − ¯Y , ˜Y = √1
2( ¯Y − ¯Z). This corresponds, in the conventions of [1], to
To compute such three-point functions using the hexagon, we first collect all the scalar excitations to one of the edges by performing the mirror transformation γ several times [1]. (See figure 2.1 for the configuration of the excitations before performing the mirror transformations.) After collecting them on the second edge (O2) on the left hexagon and
on the first edge (O1) on the right hexagon, we obtain the hexagons with {α4γ1 , α 2γ
3 , α2} and
with { ¯α4γ2 , ¯α2γ3 , ¯α1}. There are of course several other ways to collect the excitations to one
of the edges. However, the advantage of the choice described here is that all the excitations become Y after the transformation owing to the transformation property of the excitations clarified in [1]:
Y → − ¯2γ Y , Y¯ → −Y .2γ (4.1) Then, since all the excitations are of the Y type, the hexagon form factor factorises into two-particle form factors7 h(u, v).
We also study an analogous configuration in the sl(2) sector, where O1 and O2 contain D
excitations and O3 contains ¯D excitations. The hexagon form factor for this configuration
can be computed in a similar way, namely by collecting all the excitations to the one of the edges by using the mirror transformations.
4.1
Formulation in terms of multiple contour integrals
The asymptotic part of the un-normalised structure constant with three non-BPS operators, which we denote by [C123•••], is given by a sum over the partitions of all the three sets of Bethe roots into left and right subsets, ui = αi∪ ¯αi:
[C123•••] = X
αi∪ ¯αi=u(i) 3
Y
i=1
(−1)|α1|+|α2|+|α3| w`31(α1, ¯α1) w`12(α2, ¯α2) w`23(α3, ¯α3)
× H(α1|α3|α2)H( ¯α2| ¯α3| ¯α1) (4.2)
with the splitting factors given by w`(α, ¯α) = e−ipα` h<( ¯α, α) h>(α, ¯α), h > <(u, v) ≡Y j><k h(uj, vk) . (4.3)
7Here we included the matrix part A(u, v) is included in the definition of h(u, v) for the su(2) sector, as
The hexagon form factor can be computed by performing crossing transformation on all the excitations to bring them on the same edge
H(α1|α3|α2) = phase1H(α 4γ 1 ; α 2γ 3 ; α2) (4.4) H( ¯α2| ¯α3| ¯α1) = phase2H( ¯α 4γ 2 ; ¯α 2γ 3 ; ¯α1).
A subtle point is the definition of the crossing-transformed factors. For fields from the sl(2) sector it is sufficient to change the argument x± → 1/x±. In the general case the crossing
transformation is more complicated. It is computed by going to string frame, perform the analytic continuation and transforming back to the spin frame, cf. appendix F of [1]. In general the hexagon form factor contains a matrix and a scalar part, cf. equation (2) of [1]. As we mentioned above, in the sl(2) case the matrix part of the hexagon form factors is trivial and the weights in the sum over partitions are products of scalar factors:
[C123•••]asympt= X
αi∪ ¯αi=ui
(−1)|α1|+|α3|+|α3| e−ip(α1)`31 e−ip(α2)`12 e−ip(α3)`23 h(α4γ1 , α2) h(α4γ1 , α 2γ 3 ) h(α 2γ 3 , α2) h( ¯α4γ2 , ¯α1) h( ¯α24γ, ¯α 2γ 3 ) h( ¯α 2γ 3 , ¯α1) h(α1, ¯α1)h(α2, ¯α2)h(α3, ¯α3) × phase . (4.5)
For fields from the sl(2) sector the crossing transformation is done analytically continuing x±→ 1/x± and phase= 1. For su(2) fields the phase factors are derived in in AppendixC.
The explicit forms of the hexagon amplitudes in the two sectors is given in (3.50). and the factors h(u4γ, v) and h(u2γ, v2γ) are related to h(u, v) in a simple way:
h(u4γ, v) = 1/h(v, u), h(u2γ, v2γ) =
h(u, v) for sl(2),
h(u, v) eip(u)−ip(v) for su(2) . (4.6)
The unnormalised structure constant takes the same form for su(2) and sl(2) if we define
b(u, v) =
h(u2γ, v) = σ(u, v)/A(u, v) for sl(2),
e−ip(v)h(u2γ, v) = σ(u, v) for su(2) ,
with A(u, v) = 1 − 1 x−y+ 1 −x+1y− . (4.7) so that [C123•••]asympt= X αi∪ ¯αi=ui 3 Y i=1 (−1)| ¯αi| e −p(αi)`i−1,i h(αi, ¯αi) × 1 h(α2, α1)h( ¯α1, ¯α2) b(α3, α2) b(α3, α1) b( ¯α3, ¯α1) b( ¯α3, ¯α2) . (4.8)
generalisation of (3.10): [C123•••]asympt∝ ∞ X m,n,r 1 m! n! r! I Cu1 m Y j=1 µ(z1,j) dz1,j 2π I Cu2 n Y k=1 µ(z2,k) dz2,k 2π I Cu3 r Y l=1 µ(z3,l) dz3,l 2π × h 6=(z 1, z1) h(z1, u1) × h(z1, u2) b(u3, z1) e−ip(z1)`13 (4.9) × h 6=(z 2, z2) h(z2, u2) × h(u, z2) b(z2, u3) e−ip(z2)`12 ×h 6=(z 3, z3) h(z3, u3) × 1 b(u2, z3)b(z3, u) e−ip(z3)`23 ×b(z1, z3)b(z3, z1)b(z2, z3)b(z3, z2) h(z1, z2)h(z2, z1) .
where the last line describes the interactions between different sets of variables z1 and z2.
The numerator in the last line is equal to one due to the property b(u, v)b(v, u) = 1, and thus the integration over the third set of variables z3 completely decouples. This is what is
expected, since the left and the right su(2) fields do not feel each other perturbatively. The integral (4.9) splits into three independent integrals of the type already studied in [9], if it were not for the bi-local factor entangling the groups z1 and z2 of variables.
Remarkably, in the semiclassical limit → 0 and ` finite, the integration contours for the variables z1 and z2 are at macroscopic distance and h(z1, z2)h(z2, z1) = 1 + o().
In conclusion, the asymptotic coupling constant is given in the semiclassical limit again by a product of determinants. This can be used to work out a systematic quasi-classical expansion, which is however out of the scope of this paper. Our goal here is to compute the leading term and compare it with the result obtained on the string theory side [5]. In the semiclassical limit the structure constant factorises as
[C123•••]asympt∝A1×A2×A3. (4.10)
where the integralsA1, A2 and A3, defined as
A1 = ∞ X n=0 1 n! I Cu1 n Y j=1 µ(zj) dzj 2π × h6=(z, z) h(z, u1) × h(z, u2) b(u3, z) e−ip(z)`13 (4.11) A2 = ∞ X n=0 1 n! I Cu2 n Y j=1 µ(zj) dzj 2π × h6=(z, z) h(z, u2) × h(u1, z) b(z, u3) e−ip(z)`12 A3 = ∞ X n=0 1 n! I Cu3 n Y j=1 µ(zj) dzj 2π × h6=(z, z) h(z, u3) × 1 b(u2, z)b(z, u1) e−ip(z)`23.
Neglecting the subleading factors in the product of the scalar factors, we can approximate the functionalsAk by the objects we have already computed in the previous section,
Ak ∝ ∞ X n=0 1 n! I Cuk n Y j=1 dzj Λk(zj) 2πi n Y i<j ∆(zi, zj) (4.12)
where the functions Λ1, Λ2, Λ3 assemble the local factors for the three groups of integration
variables: Λ1(z) = e−i`31p(z) h(z, u 2) h(z, u1) b(u3, z) , Λ2(z) = e−i`12p(z) h(u 1, z) h(z, u2) b(z, u3) , Λ3(z) = e−i`23p(z) h(z, u3) b(u2, z) b(z, u1) . (4.13)
The three factors in the product (4.10) are exponentially small and the exponent of the product is given by
log[C123•••]asympt = 1
(Y1+Y2+Y3+ o()) . (4.14) whereY is a contour integral of a dilogarithm
Yi = ±
I
Cui
du
2π Li2(Λi(u)), i = 1, 2, 3. (4.15) where the (+) sign is for su(2) and the (−) sign is for sl(2).
4.2
Taking the semiclassical limit, su(2)
To obtain explicit expressions, we will express the local factors Λa(u) in terms of the three
quasi-momenta (4.19). Consider first the su(2) case where b(u, v) = σ(u, v). In the leading order in we have (see Appendix B)
log h(u, v) → − i y 0 x − y + ip(x) y2 − 1 = i x0 y − x − i p(y) x2− 1,
log b(u, v) → − iy
0 1/x − y − ip(x) y2− 1 − i p(y) = ix0 1/y − x+ ip(y) x2 − 1+ ip(x) (4.16)
or, after taking the product with xj = x(uj), uj ∈ u
log h(u, u) → −i Gu(x) + i
∆ − L 2 p(x) log b(u, u) → −i Gu(1/x) − i
∆ − L 2 p(x) log h(u, u) → i Gu(x) − i ∆ − L 2 p(x), log b(u, u) → i Gu(1/x) + i ∆ − L 2 p(x), (4.17)
where the resolvent for the set u is defined by8 Gu(x) = X j x0j x − xj , X j 1 x2j − 1 = ∆ − L 2 . (4.18)
The next step is to express the measure factors (4.13) in terms of the quasi-momenta of the three operators ˜ p(1)(x) = 12∆1p(x) − Gu1(x), ˜ p(2)(x) = 12∆2p(x) − Gu2(x), (4.19) ˜ p(3)(x) = 12∆3p(x) − Gu3(x). Substituting (4.17) in (4.13) we get
Λ1(x) → exp +i˜p(2)(x) − i˜p(1)(x) + i˜p(3)(1/x) ,
Λ2(z) → exp −i˜p(1)(x) − i˜p(2)(x) − i˜p(3)(1/x) ,
Λ3(z) → exp −i˜p(3)(x) − i˜p(1)(1/x) + i˜p(2)(1/x) .
(4.20)
Using the classical Bethe equations on the cut of ˜p(1), we can change the sign of ˜p(1) in the
exponent and writeY1, eq. (4.15), as
Y1 = − I Cu1 du 2π Li2(e i ˜p(1)(x)+i ˜p(2)(x)+i ˜p(3)(1/x) ). (4.21)
Here we took into account that the contour of integration changes its orientation when deformed to the second sheet. We also change the sign of the exponents in the other two integrals using the functional equation for the dilogarithm, Li2(X−1) = −Li2(X) − π
2 6 − 1
2log 2
(−X). This again leads to a minus sign in front of the integrals: Y2 = − I Cu2 du 2π Li2(e i ˜p(1)(x)+i ˜p(2)(x)+i ˜p(3)(1/x) ) Y3 = − I Cu3 du 2π Li2(e
−i˜p(2)(1/x)+i ˜p(1)(1/x)+i ˜p(3)(x)
).
(4.22)
The final formula is
log[C123•••]asymptsu(2) = −1 I Cu1∪u2 du 2π Li2(e i ˜p(1)(x)+i ˜p(2)(x)+i ˜p(3)(1/x) ) −1 I Cu3 du 2π Li2(e
−i˜p(3)(1/x)+i ˜p(2)(x)−i ˜p(1)(x)
) + subleading in .
(4.23)
This expression gives, up to subleading o(0) terms, the exponent for the all-loop
perturba-tive structure constant for three heavy fields.
Let us interpret this expression from the point of view of the spectral curves of the three heavy states which is written in termes of the classical monodromy matrix
Ω(u) = Diag ei ˆp1(u), ei ˆp2(u), ei ˆp3(u), ei ˆp4(u)|ei ˆp1(u), ei ˆp2(u), ei ˆp3(u), ei ˆp4(u) .
The finite zone solutions in this sector are characterised by cuts between 1-4 and 2-3 sheets of the Riemann surface. The Bethe equations give boundary conditions on these cuts for the combinations ˜pL = 12(˜p1− ˜p4) and ˜pR = 12(˜p2− ˜p3), representing the quasi-momenta in
the left and in the right su(2) sectors. The spectral curve of the SO(4) sector is invariant under the inversion symmetry x ↔ 1/x, which exchanges ˜pL and ˜pR
˜
pR(x) = −˜pL(1/x). (4.24)
This allows to go from the four-sheeted Riemann surface in the u-parametrization to a two-sheet Riemann surface in the x-parametrization
˜ pR(x) = −˜p(1/x) |x|>1, p˜L(x) = ˜p(x) |x|>1. (4.25) In the notations pL,R(u) via (4.25), the unnormalised structure constant takes the form
log[C123•••]asymptsu(2) = −1 I Cu1∪u2 du 2π Li2(e i ˜p(1)L +i ˜p(2)L −i˜p(3)R ) −1 I Cu3 du 2π Li2(e i ˜p(3)R +i ˜p(2)L −i˜p(1)L ) + subleading in . (4.26)
In the strong coupling limit this expression reproduces exactly the the result of the string theory computation, eq. (2.12).
4.3
Taking the semiclassical limit, sl(2)
In the case of sl(2) fields the scalar factors h(u, v) have asymptotics, cf. appendixB,
log hsl(2)(u, v) → iy0 x − y − ip(y) x2− 1 = ix0 x − y + ip(x) y2− 1, log bsl(2)(u, v) → iy0 1/x − y + ip(y) 1 − 1/x2 = ix0 x − 1/y − ip(x) 1 − 1/y2 (4.27) which gives log h(u, u) → i Gu(x),
log h(u, u) → −iGu(1/x)
log b(u, u) → i Gu(1/x) ,
log b(u, u) → −iGu(1/x) .
Substituting in (4.13), we obtain
Λ1(x) → exp +iˆp(2)(x) − iˆp(1)(x) + iˆp(3)(1/x) ,
Λ2(z) → exp −iˆp(1)(x) − iˆp(2)(x) − iˆp(3)(1/x) ,
Λ3(z) → exp −iˆp(3)(x) − iˆp(1)(1/x) + iˆp(2)(1/x) ,
(4.29)
where ˆp(k) are the sl(2) quasi-momenta,
ˆ
p(k)(x) = 12Lkp(x) + Guk(x) , k = 1, 2, 3 . (4.30)
The rest is in complete analogy with the su(2) sector. Taking into account the opposite sign of the dilogarithm, we write it as
log[C123•••]asymptsl(2) = 1 I Cu1∪u2 du 2π Li2 h ei ˆp(1)L +i ˆp (2) L −iˆp (3) R i +1 I Cu3 du 2π Li2 h ei ˆp(3)R +i ˆp (2) L −iˆp (1) L i + subleading in . (4.31)
which is what is expected from the strong coupling result [21] in (2.14).
5
Bottom mirror excitations
The full result of the structure constant requires taking into account mirror excitations on all the three edges. The general expression is too complicated to be treated here; moreover, the interaction of mirror particles in crossed channels is affected by singularities which need careful regularisation. The simplest, tractable case of mirror contribution is that of the structure constant with only one non-BPS operatos, in the channel opposed to the on the opposite edge of the physical excitations, or bottom channel, as shown in figure2.3. These mirror particles do not enter the sum over partitions and they do not interact with the other mirror excitations, so they can be factorised out and considered separately. Written schematically in terms of the fundamental excitations, the integrand is given by [25]
µ(wγ) eip(wγ)`BT (wγ) h6=(wγ, wγ) h(u, w−3γ) , (5.1) with `B = 12(L2+ L3− L1) the length of the bottom bridge of the correlator, opposed to the
operator O1. The last factor can also be transformed to the same mirror dynamics by using
(4.6), h(u, w−3γ) = 1/h(wγ, u). The mirror transformation γ is defined as the analytical continuation through the branch cut of the variable x+, namely x+ → 1/x+, as shown in
5.8.
In [1] the contribution of a single mirror particle was analysed, and shown to reproduce lowest order contribution of the expected strong coupling answer [5, 21]. The full mirror
corrections involve all the bound states, and in the integrand (2.5) all the quantities should be replaced by their bound state counterparts. Here we are able to sum all the bound state contribution, in the strong coupling limit, and to retrieve part of the strong coupling result. This imply summing over all the configurations ~n = {n1, n2, . . . }, where na is the number
of bound states of a magnons,
[C•◦◦]bottom =X ~ n B[~n] Q ana! . (5.2)
The contribution of the configuration ~n is given by9
B[~n] = (−1)n Z ∞ −∞ Y a na Y j=1 dza j 2πµ γ a(z a j) g γ a(z a j) T γ a(z a j) (5.3) × Y a 1≤i<j≤na Haaγ (zia, zja) Y a<b 1≤i≤na 1≤j≤nb Habγ(zia, zbj) , n =X a naa.
The integration contour is along the real axis in the mirror regime shown in5.8. The bi-local factors Habγ(zia, zjb) coupling two bound states of length a and b are given by
Habγ(u, v) ≡ hab(uγ, vγ) hba(vγ, uγ) , (5.4)
where hab(u, v) is the bound state counterpart of h(u, v) and is defined in (5.7). The functions
gγ
a(u) ≡ ga(uγ) and µγa(u) ≡ µa(uγ) are mirror transforms respectively of the local weight
factor ga(u) and the measure µa(u) defined as
ga(u) = eipa(u) `B ha,1(u, u) , µa(u) = 1 a (1 − 1/x[−a]x[+a])2
(1 − 1/x[+a]x[+a]) (1 − 1/x[−a]x[−a]). (5.5)
Throughout this chapter we are using the notation x[k] = x(u + ik/2). We want to take the semiclassical limit of (5.2) and (5.3), focusing on the strong coupling limit g → ∞. Since the su(2) and the sl(2) cases are treated in almost identical way, we will focus on the su(2) case and will briefly summarise the sl(2) case at the end.
5.1
Quantities for bound states at strong coupling
In this section, we determine the strong coupling limit expressions of the various quantities in the integrand (5.3). The bound state counterparts can be obtained by fusing the corre-sponding fundamental quantities. Notice that when we perform strong coupling expansion,
there are different regimes in the complex x-plane. Since the integration contours for the rapidities z are situated on the real axis in the mirror dynamics, it is enough to analyse the strong coupling limit in this regime [32]. The near-flat-space regime, where u is situated close to the singularities x(u) = ±1 is not relevant for this case, as we are concerned with semiclassical strings. We have mainly to check the mirror giant magnon regime |x(u)| > 1 and the mirror BMN regime |x(u)| = 110. The integrals over mirror particles contain the factor
e−Ea(u)`B ∼ 1
x2`B , |x(u)| > 1 (5.6)
which strongly suppresses the contribution of the mirror giant magnon particles for large values of the bridge length `B. We are therefore going to concentrate on the BMN mirror
regime |x| = 1. -2g' 2g' x cut[+a] physical regime mirror regime [-a] x cut
Figure 5.8: The rule for analytic continuation from the BMN mirror regime to BMN physical regime at strong coupling, when the real axis is pinched between the branch cuts of the Zhukovsky variables x+ and x−.
As illustrated in5.8, the contributions from the mirror BMN regime at strong coupling can be determined by first taking the strong-coupling limit of the relevant quantities in the physical regime and then analytically continuing them to the lower half of the unit circle |x| = 1. This simple rule should be applied with care for the bound-state quantities, which may have an array of branch cuts. In this case, the passage to the mirror regime of an object associated to a bound state of size a is done by substituting x[+a](u) by 1/x[+a](u), that is by analytically continuing u through the branch cut of the Zhukovsky variable x[+a](u) and leaving the other cut untouched.
10The denomination of the various regimes follows the analytical continuation of the corresponding ones
In the strong coupling limit, the different branch cuts collapse on each other and on the real axis and the dependence on the rapidities will be given by variable with a single branch cut x(u). In the mirror giant magnon regime, x[+a] ' x[−a] ' x, while in the mirror BMN
regime, where the branch cut is situated, x[+a] ' 1/x[−a] → x. The net result is that after
both mirror transformation and strong coupling limit, x[±a] → 1/x, which is equivalent to
continuing x to the lower half unit circle U−.
Special care has to be devoted to the continuation of the dressing phase for bound states to the mirror dynamics, where extra cuts appear. As we explain in appendix D based on [41, 42], the dressing phase appears in combination with other functions which cancel exactly the cuts on the real axis and all the other cuts between those of x[−a](u) and x[+a](u).
The same combination has no branch cut below that of x[+a](u) in the physical dynamics,
therefore we are again in the situation represented in figure5.8and we can use the analytical continuation from the BMN physical regime to the BMN mirror regime.
The scalar factor Hab(u, v). The scalar factor for scattering of two bound states of length
a and b is given by hab(u, v) = a−1 2 Y k=−a−12 b−1 2 Y l=−b−12 h(u[2k], v[2l]) . (5.7)
The symmetric scalar factor Hab(u, v) = hab(u, v)hba(v, u) is then given by
Hab(u, v) = x[−a]− y[−b] x[+a]− y[−b] x[+a]− y[+b] x[−a]− y[+b] 1 − 1/x[−a]y[+b] 1 − 1/x[+a]y[+b] 1 − 1/x[+a]y[−b] 1 − 1/x[−a]y[−b] . (5.8)
The dressing factor dropped out from the expression of the symmetric factor. In the strong coupling limit in the mirror dynamics Hγ(u, v) takes the simple form
Habγ(u, v) ' u − v − i a−b 2 u − v − ia+b2 u − v + ia−b2 u − v + ia+b2 . (5.9) We notice that the pairwise interaction takes in the strong coupling limit the same form as the interaction of the bound states (3.19) in the asymptotic structure constant,
Habγ(u, v) ' ∆ab(u[−a], v[−b]) = ∆ab(u −12ia, v − 12ib) , (5.10)
the only difference being that the position of the pole is shifted to v = u ± i(a + b)/2.
The measure µa(u). The expression for the measure for a bound state, eq.(5.5), is
µa(u) =
1 a
(1 − 1/x[−a]x[+a])2
Performing mirror transformation for µa(u) and expanding at strong coupling in the mirror
BMN regime |x| = 1, we find
µa(uγ) '
1
a. (5.12)
The factor ga(u). Recall that
ga(u) = eipa(u)`B ha1(u, u) , (5.13) with ha1(u, v) = u[−a]− v− u[+a]− v− 1 − 1/x[−a]y+ 1 − 1/x[+a]y+ 1 − 1/x[+a]y− 1 − 1/x[−a]y− 1 σa,1 . (5.14) After the continuation to the mirror dynamics, σa,1(uγ, v) has extra cuts between those
situated at u − ia/2 and u + ia/2 with u ∈ [−2g0, 2g0]. In particular, for even a one of those cuts is situated on the real axis, i.e. on the contour of integration for the mirror particle contribution. These cuts are compensated by an extra factor coming from the normalisation of the transfer matrix matrix, as we will show below. The quantity we have to consider is ˜ ga = ga(u) R(−)[2−a] R(+)[2−a]. . . R(−)[a] R(+)[a] . (5.15)
Here and below we use the notation
R(±)(u) = (x − x∓) ≡Y j (x(u) − x∓(uj)), B(±)(u) = (1/x − x∓) ≡Y j (1/x(u) − x∓(uj)). (5.16)
where the functions R(±)(u), B(±)(u) play the role of the Baxter polynomials in the Zhukovsky
plane and encode the rapidities of the incoming state. The simplest strategy to take the strong coupling limit of the quantity above is to compute it first in the BMN physical dynamics and the analytically continue to the BMN mirror dynamics. This can be done because there is no singularity to be met on the path of the analytical continuation. Taking the strong coupling limit, one can express ga(u) in terms of the three quasi-momenta (3.49)
ga(u) → eiap(x)`B+iaG(x)−ia ∆−L1
2 p(x) = eia( ˜p2(x)+ ˜p3(x)− ˜p1(x)) (5.17)
and, after the continuation to the mirror regime one has ˜
ga(uγ) = → eiap(1/x)`B−ia ∆−L1
